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In electromagnetism, there are two kinds of **dipoles**:

**Electromagnetism** is a branch of physics involving the study of the **electromagnetic force**, a type of physical interaction that occurs between electrically charged particles. The electromagnetic force is carried by electromagnetic fields composed of electric fields and magnetic fields, is responsible for electromagnetic radiation such as light, and is one of the four fundamental interactions in nature. The other three fundamental interactions are the strong interaction, the weak interaction, and gravitation. At high energy the weak force and electromagnetic force are unified as a single electroweak force.

- Classification
- Molecular dipoles
- Quantum mechanical dipole operator
- Atomic dipoles
- Field of a static magnetic dipole
- Magnitude
- Vector form
- Magnetic vector potential
- Field from an electric dipole
- Torque on a dipole
- Dipole radiation
- See also
- Notes
- References
- External links

- An electric dipole deals with the separation of the positive and negative charges found in any electromagnetic system. A simple example of this system is a pair of electric charges of equal magnitude but opposite sign separated by some typically small distance. (A permanent electric dipole is called an electret.)
- A magnetic dipole is the closed circulation of an electric current system. A simple example is a single loop of wire with constant current through it. A bar magnet is an example of a magnet with a permanent magnetic dipole moment.
^{ [1] }^{ [2] }

The **electric dipole moment** is a measure of the separation of positive and negative electrical charges within a system, that is, a measure of the system's overall polarity. The SI units for electric dipole moment are coulomb-meter (C⋅m); however, a commonly used unit in atomic physics and chemistry is the debye (D).

**Electric charge** is the physical property of matter that causes it to experience a force when placed in an electromagnetic field. There are two types of electric charge: *positive* and *negative*. Like charges repel and unlike attract. An object with an absence of net charge is referred to as *neutral*. Early knowledge of how charged substances interact is now called classical electrodynamics, and is still accurate for problems that do not require consideration of quantum effects.

An **electret** is a dielectric material that has a quasi-permanent electric charge or dipole polarisation. An electret generates internal and external electric fields, and is the electrostatic equivalent of a permanent magnet. Although Oliver Heaviside coined this term in 1885, materials with electret properties were already known to science and had been studied since the early 1700s. One particular example is the electrophorus, a device consisting of a slab with electret properties and a separate metal plate. The electrophorus was originally invented by Johan Carl Wilcke in Sweden and again by Alessandro Volta in Italy.

Whether electric or magnetic, dipoles can be characterized by their dipole moment, a vector quantity. For the simple electric dipole, the electric dipole moment points from the negative charge towards the positive charge, and has a magnitude equal to the strength of each charge times the separation between the charges. (To be precise: for the definition of the dipole moment, one should always consider the "dipole limit", where, for example, the distance of the generating charges should *converge* to 0 while simultaneously, the charge strength should *diverge* to infinity in such a way that the product remains a positive constant.)

For the magnetic (dipole) current loop, the magnetic dipole moment points through the loop (according to the right hand grip rule), with a magnitude equal to the current in the loop times the area of the loop.

Similar to magnetic current loops, the electron particle and some other fundamental particles have magnetic dipole moments, as an electron generates a magnetic field identical to that generated by a very small current loop. However, an electron's magnetic dipole moment is not due to a current loop, but to an intrinsic property of the electron.^{ [3] } The electron may also have an *electric* dipole moment though such has yet to be observed (see electron electric dipole moment).

The **electron** is a subatomic particle, symbol ^{}e^{−}_{} or ^{}β^{−}_{}, whose electric charge is negative one elementary charge. Electrons belong to the first generation of the lepton particle family, and are generally thought to be elementary particles because they have no known components or substructure. The electron has a mass that is approximately 1/1836 that of the proton. Quantum mechanical properties of the electron include an intrinsic angular momentum (spin) of a half-integer value, expressed in units of the reduced Planck constant, *ħ*. Being fermions, no two electrons can occupy the same quantum state, in accordance with the Pauli exclusion principle. Like all elementary particles, electrons exhibit properties of both particles and waves: they can collide with other particles and can be diffracted like light. The wave properties of electrons are easier to observe with experiments than those of other particles like neutrons and protons because electrons have a lower mass and hence a longer de Broglie wavelength for a given energy.

A **magnetic field** is a vector field that describes the magnetic influence of electric charges in relative motion and magnetized materials. The effects of magnetic fields are commonly seen in permanent magnets, which pull on magnetic materials and attract or repel other magnets. Magnetic fields surround and are created by magnetized material and by moving electric charges such as those used in electromagnets. They exert forces on nearby moving electrical charges and torques on nearby magnets. In addition, a magnetic field that varies with location exerts a force on magnetic materials. Both the strength and direction of a magnetic field vary with location. As such, it is described mathematically as a vector field.

In science and engineering, an **intrinsic** property is a property of a specified subject that exists itself or within the subject. An **extrinsic** property is not essential or inherent to the subject that is being characterized. For example, density is an intrinsic property of any physical object, whereas weight is an extrinsic property that depends on another object.

A permanent magnet, such as a bar magnet, owes its magnetism to the intrinsic magnetic dipole moment of the electron. The two ends of a bar magnet are referred to as poles—not to be confused with monopoles, see Classification below)—and may be labeled "north" and "south". In terms of the Earth's magnetic field, they are respectively "north-seeking" and "south-seeking" poles: if the magnet were freely suspended in the Earth's magnetic field, the north-seeking pole would point towards the north and the south-seeking pole would point towards the south. The dipole moment of the bar magnet points from its magnetic south to its magnetic north pole. In a magnetic compass, the north pole of a bar magnet points north. However, that means that Earth's geomagnetic north pole is the *south* pole (south-seeking pole) of its dipole moment and vice versa.

In particle physics, a **magnetic monopole** is a hypothetical elementary particle that is an isolated magnet with only one magnetic pole. A magnetic monopole would have a net "magnetic charge". Modern interest in the concept stems from particle theories, notably the grand unified and superstring theories, which predict their existence.

A **compass** is an instrument used for navigation and orientation that shows direction relative to the geographic cardinal directions. Usually, a diagram called a compass rose shows the directions north, south, east, and west on the compass face as abbreviated initials. When the compass is used, the rose can be aligned with the corresponding geographic directions; for example, the "N" mark on the rose points northward. Compasses often display markings for angles in degrees in addition to the rose. North corresponds to 0°, and the angles increase clockwise, so east is 90° degrees, south is 180°, and west is 270°. These numbers allow the compass to show magnetic North azimuths or true North azimuths or bearings, which are commonly stated in this notation. If magnetic declination between the magnetic North and true North at latitude angle and longitude angle is known, then direction of magnetic North also gives direction of true North.

The only known mechanisms for the creation of magnetic dipoles are by current loops or quantum-mechanical spin since the existence of magnetic monopoles has never been experimentally demonstrated.

The term comes from the Greek δίς (*dis*), "twice"^{ [4] } and πόλος (*polos*), "axis".^{ [5] }^{ [6] }

A *physical dipole* consists of two equal and opposite point charges: in the literal sense, two poles. Its field at large distances (i.e., distances large in comparison to the separation of the poles) depends almost entirely on the dipole moment as defined above. A *point (electric) dipole* is the limit obtained by letting the separation tend to 0 while keeping the dipole moment fixed. The field of a point dipole has a particularly simple form, and the order-1 term in the multipole expansion is precisely the point dipole field.

Although there are no known magnetic monopoles in nature, there are magnetic dipoles in the form of the quantum-mechanical spin associated with particles such as electrons (although the accurate description of such effects falls outside of classical electromagnetism). A theoretical magnetic *point dipole* has a magnetic field of exactly the same form as the electric field of an electric point dipole. A very small current-carrying loop is approximately a magnetic point dipole; the magnetic dipole moment of such a loop is the product of the current flowing in the loop and the (vector) area of the loop.

Any configuration of charges or currents has a 'dipole moment', which describes the dipole whose field is the best approximation, at large distances, to that of the given configuration. This is simply one term in the multipole expansion when the total charge ("monopole moment") is 0—as it *always* is for the magnetic case, since there are no magnetic monopoles. The dipole term is the dominant one at large distances: Its field falls off in proportion to 1/*r*^{3}, as compared to 1/*r*^{4} for the next (quadrupole) term and higher powers of 1/*r* for higher terms, or 1/*r*^{2} for the monopole term.

Many molecules have such dipole moments due to non-uniform distributions of positive and negative charges on the various atoms. Such is the case with polar compounds like hydrogen fluoride (HF), where electron density is shared unequally between atoms. Therefore, a molecule's dipole is an electric dipole with an inherent electric field that should not be confused with a magnetic dipole which generates a magnetic field.

The physical chemist Peter J. W. Debye was the first scientist to study molecular dipoles extensively, and, as a consequence, dipole moments are measured in units named * debye * in his honor.

For molecules there are three types of dipoles:

- Permanent dipoles
- These occur when two atoms in a molecule have substantially different electronegativity: One atom attracts electrons more than another, becoming more negative, while the other atom becomes more positive. A molecule with a permanent dipole moment is called a
*polar*molecule. See dipole–dipole attractions. - Instantaneous dipoles
- These occur due to chance when electrons happen to be more concentrated in one place than another in a molecule, creating a temporary dipole. These dipoles are smaller in magnitude than permanent dipoles, but still play a large role in chemistry and biochemistry due to their prevalence. See instantaneous dipole.
- Induced dipoles
- These can occur when one molecule with a permanent dipole repels another molecule's electrons,
*inducing*a dipole moment in that molecule. A molecule is*polarized*when it carries an induced dipole. See induced-dipole attraction.

More generally, an induced dipole of *any* polarizable charge distribution *ρ* (remember that a molecule has a charge distribution) is caused by an electric field external to *ρ*. This field may, for instance, originate from an ion or polar molecule in the vicinity of *ρ* or may be macroscopic (e.g., a molecule between the plates of a charged capacitor). The size of the induced dipole moment is equal to the product of the strength of the external field and the dipole polarizability of *ρ*.

Dipole moment values can be obtained from measurement of the dielectric constant. Some typical gas phase values in debye units are:^{ [7] }

- carbon dioxide: 0
- carbon monoxide: 0.112 D
- ozone: 0.53 D
- phosgene: 1.17 D
- water vapor: 1.85 D
- hydrogen cyanide: 2.98 D
- cyanamide: 4.27 D
- potassium bromide: 10.41 D

Potassium bromide (KBr) has one of the highest dipole moments because it is an ionic compound that exists as a molecule in the gas phase.

The overall dipole moment of a molecule may be approximated as a vector sum of bond dipole moments. As a vector sum it depends on the relative orientation of the bonds, so that from the dipole moment information can be deduced about the molecular geometry.

For example, the zero dipole of CO_{2} implies that the two C=O bond dipole moments cancel so that the molecule must be linear. For H_{2}O the O−H bond moments do not cancel because the molecule is bent. For ozone (O_{3}) which is also a bent molecule, the bond dipole moments are not zero even though the O−O bonds are between similar atoms. This agrees with the Lewis structures for the resonance forms of ozone which show a positive charge on the central oxygen atom.

An example in organic chemistry of the role of geometry in determining dipole moment is the *cis* and *trans* isomers of 1,2-dichloroethene. In the *cis* isomer the two polar C−Cl bonds are on the same side of the C=C double bond and the molecular dipole moment is 1.90 D. In the *trans* isomer, the dipole moment is zero because the two C−Cl bonds are on opposite sides of the C=C and cancel (and the two bond moments for the much less polar C−H bonds also cancel).

Another example of the role of molecular geometry is boron trifluoride, which has three polar bonds with a difference in electronegativity greater than the traditionally cited threshold of 1.7 for ionic bonding. However, due to the equilateral triangular distribution of the fluoride ions about the boron cation center, the molecule **as a whole** does not exhibit any identifiable pole: one cannot construct a plane that divides the molecule into a net negative part and a net positive part.

Consider a collection of *N* particles with charges *q _{i}* and position vectors

Notice that this definition is valid only for neutral atoms or molecules, i.e. total charge equal to zero. In the ionized case, we have

where is the center of mass of the molecule/group of particles.^{ [8] }

A non-degenerate (*S*-state) atom can have only a zero permanent dipole. This fact follows quantum mechanically from the inversion symmetry of atoms. All 3 components of the dipole operator are antisymmetric under inversion with respect to the nucleus,

where is the dipole operator and is the inversion operator.

The permanent dipole moment of an atom in a non-degenerate state (see degenerate energy level) is given as the expectation (average) value of the dipole operator,

where is an *S*-state, non-degenerate, wavefunction, which is symmetric or antisymmetric under inversion: . Since the product of the wavefunction (in the ket) and its complex conjugate (in the bra) is always symmetric under inversion and its inverse,

it follows that the expectation value changes sign under inversion. We used here the fact that , being a symmetry operator, is unitary: and by definition the Hermitian adjoint may be moved from bra to ket and then becomes . Since the only quantity that is equal to minus itself is the zero, the expectation value vanishes,

In the case of open-shell atoms with degenerate energy levels, one could define a dipole moment by the aid of the first-order Stark effect. This gives a non-vanishing dipole (by definition proportional to a non-vanishing first-order Stark shift) only if some of the wavefunctions belonging to the degenerate energies have opposite parity; i.e., have different behavior under inversion. This is a rare occurrence, but happens for the excited H-atom, where 2s and 2p states are "accidentally" degenerate (see article Laplace–Runge–Lenz vector for the origin of this degeneracy) and have opposite parity (2s is even and 2p is odd).

The far-field strength, *B*, of a dipole magnetic field is given by

where

*B*is the strength of the field, measured in teslas*r*is the distance from the center, measured in metres*λ*is the magnetic latitude (equal to 90° −*θ*) where*θ*is the magnetic colatitude, measured in radians or degrees from the dipole axis^{ [note 1] }*m*is the dipole moment, measured in ampere-square metres or joules per tesla*μ*_{0}is the permeability of free space, measured in henries per metre.

Conversion to cylindrical coordinates is achieved using *r*^{2} = *z*^{2} + *ρ*^{2} and

where *ρ* is the perpendicular distance from the *z*-axis. Then,

The field itself is a vector quantity:

where

**B**is the field**r**is the vector from the position of the dipole to the position where the field is being measured*r*is the absolute value of**r**: the distance from the dipole**r̂**=**r**/*r*is the unit vector parallel to**r**;**m**is the (vector) dipole moment*μ*_{0}is the permeability of free space

This is *exactly* the field of a point dipole, *exactly* the dipole term in the multipole expansion of an arbitrary field, and *approximately* the field of any dipole-like configuration at large distances.

The vector potential **A** of a magnetic dipole is

with the same definitions as above.

The electrostatic potential at position **r** due to an electric dipole at the origin is given by:

where **p** is the (vector) dipole moment, and *є*_{0} is the permittivity of free space.

This term appears as the second term in the multipole expansion of an arbitrary electrostatic potential Φ(**r**). If the source of Φ(**r**) is a dipole, as it is assumed here, this term is the only non-vanishing term in the multipole expansion of Φ(**r**). The electric field from a dipole can be found from the gradient of this potential:

This is formally identical to the expression for the magnetic field of a point magnetic dipole with only a few names changed. In a real dipole, however, where the charges are physically separate, the "internal" field lines are different, since the magnetic field lines are continuous, while those of the electric field diverge or converge from the point charges. For further discussions about the internal field of dipoles, see^{ [2] }^{ [9] } or Magnetic moment#Internal magnetic field of a dipole.

Since the direction of an electric field is defined as the direction of the force on a positive charge, electric field lines point away from a positive charge and toward a negative charge.

When placed in an homogeneous electric or magnetic field, equal but opposite forces arise on each side of the dipole creating a torque **τ**}:

for an electric dipole moment **p** (in coulomb-meters), or

for a magnetic dipole moment **m** (in ampere-square meters).

The resulting torque will tend to align the dipole with the applied field, which in the case of an electric dipole, yields a potential energy of

- .

The energy of a magnetic dipole is similarly

- .

In addition to dipoles in electrostatics, it is also common to consider an electric or magnetic dipole that is oscillating in time. It is an extension, or a more physical next-step, to spherical wave radiation.

In particular, consider a harmonically oscillating electric dipole, with angular frequency *ω* and a dipole moment *p*_{0} along the **ẑ** direction of the form

In vacuum, the exact field produced by this oscillating dipole can be derived using the retarded potential formulation as:

For *rω*/*c* ≫ 1, the far-field takes the simpler form of a radiating "spherical" wave, but with angular dependence embedded in the cross-product:^{ [10] }

The time-averaged Poynting vector

is not distributed isotropically, but concentrated around the directions lying perpendicular to the dipole moment, as a result of the non-spherical electric and magnetic waves. In fact, the spherical harmonic function (sin *θ*) responsible for such toroidal angular distribution is precisely the *l* = 1 "p" wave.

The total time-average power radiated by the field can then be derived from the Poynting vector as

Notice that the dependence of the power on the fourth power of the frequency of the radiation is in accordance with the Rayleigh scattering, and the underlying effects why the sky consists of mainly blue colour.

A circular polarized dipole is described as a superposition of two linear dipoles.

- Polarization density
- Magnetic dipole models
- Dipole model of the Earth's magnetic field
- Electret
- Indian Ocean Dipole and Subtropical Indian Ocean Dipole, two oceanographic phenomena
- Magnetic dipole-dipole interaction
- Spin magnetic moment
- Monopole
- Solid harmonics
- Axial multipole moments
- Cylindrical multipole moments
- Spherical multipole moments
- Laplace expansion
- Molecular solid
- Magnetic moment#Internal magnetic field of a dipole

- ↑ Magnetic colatitude is 0 along the dipole's axis and 90° in the plane perpendicular to its axis.

In quantum mechanics, a **Hamiltonian** is an operator corresponding to the sum of the kinetic energies plus the potential energies for all the particles in the system. It is usually denoted by , but also or to highlight its function as an operator. Its spectrum is the set of possible outcomes when one measures the total energy of a system. Because of its close relation to the time-evolution of a system, it is of fundamental importance in most formulations of quantum theory.

A **magnetic dipole** is the limit of either a closed loop of electric current or a pair of poles as the size of the source is reduced to zero while keeping the magnetic moment constant. It is a magnetic analogue of the electric dipole, but the analogy is not perfect. In particular, a magnetic monopole, the magnetic analogue of an electric charge, has never been observed. Moreover, one form of magnetic dipole moment is associated with a fundamental quantum property—the spin of elementary particles.

**Synchrotron radiation** is the electromagnetic radiation emitted when charged particles are accelerated radially, e.g., when they are subject to an acceleration perpendicular to their velocity. It is produced, for example, in synchrotrons using bending magnets, undulators and/or wigglers. If the particle is non-relativistic, then the emission is called cyclotron emission. If, on the other hand, the particles are relativistic, sometimes referred to as ultrarelativistic, the emission is called synchrotron emission. Synchrotron radiation may be achieved artificially in synchrotrons or storage rings, or naturally by fast electrons moving through magnetic fields. The radiation produced in this way has a characteristic polarization and the frequencies generated can range over the entire electromagnetic spectrum which is also called continuum radiation.

In atomic physics, **hyperfine structure** comprises small shifts and splittings in the energy levels of atoms, molecules, and ions, due to interaction between the state of the nucleus and the state of the electron clouds.

The **magnetic moment** is the magnetic strength and orientation of a magnet or other object that produces a magnetic field. Examples of objects that have magnetic moments include: loops of electric current, permanent magnets, elementary particles, various molecules, and many astronomical objects.

In physics, the **Rabi cycle** is the cyclic behaviour of a two-level quantum system in the presence of an oscillatory driving field. A great variety of physical processes belonging to the areas of quantum computing, condensed matter, atomic and molecular physics, and nuclear and particle physics can be conveniently studied in terms of two-level quantum mechanical systems, and exhibit Rabi flopping when coupled to an oscillatory driving field. The effect is important in quantum optics, magnetic resonance and quantum computing, and is named after Isidor Isaac Rabi.

In electromagnetism, **permeability** is the measure of the ability of a material to support the formation of a magnetic field within itself, otherwise known as distributed inductance in transmission line theory. Hence, it is the degree of magnetization that a material obtains in response to an applied magnetic field. Magnetic permeability is typically represented by the (italicized) Greek letter *μ*. The term was coined in September 1885 by Oliver Heaviside. The reciprocal of magnetic permeability is magnetic reluctivity.

The term **magnetic potential** can be used for either of two quantities in classical electromagnetism: the *magnetic vector potential*, or simply *vector potential*, **A**; and the *magnetic scalar potential**ψ*. Both quantities can be used in certain circumstances to calculate the magnetic field **B**.

In classical electromagnetism, **magnetization** or **magnetic polarization** is the vector field that expresses the density of permanent or induced magnetic dipole moments in a magnetic material. The origin of the magnetic moments responsible for magnetization can be either microscopic electric currents resulting from the motion of electrons in atoms, or the spin of the electrons or the nuclei. Net magnetization results from the response of a material to an external magnetic field, together with any unbalanced magnetic dipole moments that may be inherent in the material itself; for example, in ferromagnets. Magnetization is not always uniform within a body, but rather varies between different points. Magnetization also describes how a material responds to an applied magnetic field as well as the way the material changes the magnetic field, and can be used to calculate the forces that result from those interactions. It can be compared to electric polarization, which is the measure of the corresponding response of a material to an electric field in electrostatics. Physicists and engineers usually define magnetization as the quantity of magnetic moment per unit volume. It is represented by a pseudovector **M**.

In many-body theory, the term **Green's function** is sometimes used interchangeably with correlation function, but refers specifically to correlators of field operators or creation greenand annihilation operators.

**Magnetic dipole–dipole interaction**, also called **dipolar coupling**, refers to the direct interaction between two magnetic dipoles.

The **transition dipole moment** or **transition moment**, usually denoted for a transition between an initial state, , and a final state, , is the electric dipole moment associated with the transition between the two states. In general the transition dipole moment is a complex vector quantity that includes the phase factors associated with the two states. Its direction gives the polarization of the transition, which determines how the system will interact with an electromagnetic wave of a given polarization, while the square of the magnitude gives the strength of the interaction due to the distribution of charge within the system. The SI unit of the transition dipole moment is the Coulomb-meter (Cm); a more conveniently sized unit is the Debye (D).

**Resonance fluorescence** is the process in which a two-level atom system interacts with the quantum electromagnetic field if the field is driven at a frequency near to the natural frequency of the atom.

An LC circuit can be quantized using the same methods as for the quantum harmonic oscillator. An **LC circuit** is a variety of resonant circuit, and consists of an inductor, represented by the letter L, and a capacitor, represented by the letter C. When connected together, an electric current can alternate between them at the circuit's resonant frequency:

In quantum field theory, and especially in quantum electrodynamics, the interacting theory leads to infinite quantities that have to be absorbed in a renormalization procedure, in order to be able to predict measurable quantities. The renormalization scheme can depend on the type of particles that are being considered. For particles that can travel asymptotically large distances, or for low energy processes, the **on-shell scheme**, also known as the physical scheme, is appropriate. If these conditions are not fulfilled, one can turn to other schemes, like the Minimal subtraction scheme.

**Multipole radiation** is a theoretical framework for the description of electromagnetic or gravitational radiation from time-dependent distributions of distant sources. These tools are applied to physical phenomena which occur at a variety of length scales - from gravitational waves due to galaxy collisions to gamma radiation resulting from nuclear decay. Multipole radiation is analyzed using similar multipole expansion techniques that describe fields from static sources, however there are important differences in the details of the analysis because multipole radiation fields behave quite differently from static fields. This article is primarily concerned with electromagnetic multipole radiation, although the treatment of gravitational waves is similar.

**Magnetic resonance** is a phenomenon in quantum mechanics that affects a magnetic dipole when placed in a uniform static magnetic field. Its energy is split into a finite number of energy levels, depending on the value of quantum number of angular momentum. This is similar to energy quantization for atoms, say ^{}e^{−}_{} in H atom; in this case the atom, in interaction to an external electric field, transitions between different energy levels by absorbing or emitting photons. Similarly if a magnetic dipole is perturbed with electromagnetic field of proper frequency( ), it can transit between its energy eigenstates, but as the separation between energy eigenvalues is small, the frequency of the photon will be the microwave or radio frequency range. If the dipole is tickled with a field of another frequency, it is unlikely to transition. This phenomenon is similar to what occurs when a system is acted on by a periodic force of frequency equal to its natural frequency.

**Magnetic current** is, nominally, a current composed of fictitious moving magnetic monopoles. It has the dimensions of volts. The usual symbol for magnetic current is which is analogous to for electric current. Magnetic currents produce an electric field analogously to the production of a magnetic field by electric currents. **Magnetic current density**, which has the units of V/m², is usually represented by the symbols and . The superscripts indicate total and impressed magnetic current density. The impressed currents are the energy sources. In many useful cases, a distribution of electric charge can be mathematically replaced by an equivalent distribution of magnetic current. This artifice can be used to simplify some electromagnetic field problems. It is possible to use both electric current densities and magnetic current densities in the same analysis.

**Hertz vectors**, or the **Hertz vector potentials**, are an alternative formulation of the electromagnetic potentials. They are most often introduced in electromagnetic theory textbooks as practice problems for students to solve. There are multiple cases where they have a practical use, including antennas and waveguides. Though they are sometimes used in such practice problems, they are still rarely mentioned in most electromagnetic theory courses, and when they are they are often not practiced in a manner that demonstrates when they may be useful or provide a simpler method to solving a problem than more commonly practiced methods.

- ↑ Brau, Charles A. (2004).
*Modern Problems in Classical Electrodynamics*. Oxford University Press. ISBN 0-19-514665-4. - 1 2 Griffiths, David J. (1999).
*Introduction to Electrodynamics*(3rd ed.). Prentice Hall. ISBN 0-13-805326-X. - ↑ Griffiths, David J. (1994).
*Introduction to Quantum Mechanics*. Prentice Hall. ISBN 978-0-13-124405-4. - ↑ δίς, Henry George Liddell, Robert Scott,
*A Greek-English Lexicon*, on Perseus - ↑ πόλος, Henry George Liddell, Robert Scott,
*A Greek-English Lexicon*, on Perseus - ↑ "dipole, n.".
*Oxford English Dictionary*(second ed.). Oxford University Press. 1989. - ↑ Weast, Robert C. (1984).
*CRC Handbook of Chemistry and Physics*(65th ed.). CRC Press. ISBN 0-8493-0465-2. - ↑ http://www.av8n.com/physics/electric-dipole.htm#eq-dipole-ref
- ↑ Jackson, John D. (1999).
*Classical Electrodynamics, 3rd Ed*. Wiley. pp. 148–150. ISBN 978-0-471-30932-1. - ↑ David J. Griffiths, Introduction to Electrodynamics, Prentice Hall, 1999, page 447

- USGS Geomagnetism Program
- Fields of Force: a chapter from an online textbook
- Electric Dipole Potential by Stephen Wolfram and Energy Density of a Magnetic Dipole by Franz Krafft. Wolfram Demonstrations Project.

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